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How the site degree influences quantum probability on inhomogeneous substrates

A. M. C. Souza¹, R. F. S. Andrade², N. A. M. Araújo^3,4, A. Vezzani^5,6, and H. J. Herrmann^7,8

¹Departamento de Física, Universidade Federal de Sergipe, 49100-000 Sao Cristovao, Brazil
²Instituto de Física, Universidade Federal da Bahia, 40210-210 Salvador, Brazil
³Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, P-1749-016 Lisboa, Portugal
⁴Centro de Física Teórica e Computacional, Universidade de Lisboa, P-1749-003 Lisboa, Portugal
⁵IMEM-CNR, Parco Area delle Scienze, 37/A-43124 Parma, Italy
⁶Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, via G.P. Usberti, 7/A-43124 Parma, Italy
⁷Computational Physics, IfB, ETH-Hönggerberg, Schafmattstrasse 6, 8093 Zürich, Switzerland
⁸Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza, Brazil

Phys. Rev. E 95, 042130 – Published 18 April, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.042130

Abstract

We investigate the effect of the node degree and energy $E$ on the electronic wave function for regular and irregular structures, namely, regular lattices, disordered percolation clusters, and complex networks. We evaluate the dependency of the quantum probability for each site on its degree. For a class of biregular structures formed by two disjoint subsets of sites sharing the same degree, the probability $P_{k} (E)$ of finding the electron on any site with $k$ neighbors is independent of $E \neq 0$ , a consequence of an exact analytical result that we prove for any bipartite lattice. For more general nonbipartite structures, $P_{k} (E)$ may depend on $E$ as illustrated by an exact evaluation of a one-dimensional semiregular lattice: $P_{k} (E)$ is large for small values of $E$ when $k$ is also small, and its maximum values shift towards large values of $| E |$ with increasing $k$ . Numerical evaluations of $P_{k} (E)$ for two different types of percolation clusters and the Apollonian network suggest that this observed feature might be generally valid.

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How the site degree influences quantum probability on inhomogeneous substrates

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